[[198,12,7]] · QEC Challenge

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[[198,12,7]] d =

198

kd²/n

2.97

X/Z

0.0074

4.4721

layers

swaps

487

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Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 7 · witness weight 7 (claimed exact)

witness operator (support, 7 qubits)

[23, 33, 87, 114, 159, 175, 191]

d_Z 7 · witness weight 7 (claimed exact)

witness operator (support, 7 qubits)

[61, 70, 73, 86, 135, 164, 183]

certificate exact, d = 7 · scipy/HiGHS cutoff IP

X: no logical < 7 exists; Z: no logical < 7 exists

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 2–6 (mean 4.68) · H_Z 2–6 (mean 4.768)

qubit degrees H_X 1–5 (mean 2.434) · H_Z 1–4 (mean 2.288)

trapping sets H_X (1,1)×42 (2,0)×8 (3,0)×23 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 42 (1,2): 48 (1,3): 90 (1,4): 16 (1,5): 2 (2,0): 8 (2,1): 52 (2,2): 157 (2,3): 204 (2,4): 434 (2,5): 121 (2,6): 23 (2,7): 3 (3,0): 23 (3,1): 110 (3,2): 389 (3,3): 1068 (3,4): 1719 (3,5): 2720 (3,6): 1073 (3,7): 646 (3,8): 180 (3,9): 33

trapping sets H_Z (1,1)×50 (2,0)×21 (3,1)×168 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 50 (1,2): 54 (1,3): 81 (1,4): 13 (2,0): 21 (2,1): 57 (2,2): 189 (2,3): 242 (2,4): 354 (2,5): 75 (2,6): 20 (3,1): 168 (3,2): 489 (3,3): 1304 (3,4): 1916 (3,5): 2116 (3,6): 696 (3,7): 474 (3,8): 102 (3,9): 37 (3,10): 3

witness diameter X 9.8489 · Z 7.6158 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

X checkZ checkqubit site (99)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 487 nearest-neighbor SWAPs per round in total, at most 5 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @FarLab and @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction f = 1 + x + x^-2 y², g = 1 + xy + x^-2 y^-1; annulus-type boundary engine, (11,9) lattice.

model Claude Opus 4.8 (claimed, not verified)

builds on arXiv:2504.08887

date 2026-06-16

notes Distance certified exact by cutoff MILP.

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 103 (max weight 6) · Z-checks 95 (max weight 6)

H_X (103 checks, sparse supports)

[2, 18, 27, 99, 118, 128] [3, 19, 28, 100, 119, 129] [4, 20, 29, 101, 120, 130] [5, 21, 30, 102, 121, 131] [6, 22, 31, 103, 122, 132] [7, 23, 32, 104, 123, 133] [8, 24, 33, 105, 124, 134] [11, 27, 36, 108, 127, 137] [12, 28, 37, 109, 128, 138] [13, 29, 38, 110, 129, 139] [14, 30, 39, 111, 130, 140] [15, 31, 40, 112, 131, 141] [16, 32, 41, 113, 132, 142] [17, 33, 42, 114, 133, 143] [20, 36, 45, 117, 136, 146] [21, 37, 46, 118, 137, 147] [22, 38, 47, 119, 138, 148] [23, 39, 48, 120, 139, 149] [24, 40, 49, 121, 140, 150] [25, 41, 50, 122, 141, 151] [26, 42, 51, 123, 142, 152] [29, 45, 54, 126, 145, 155] [30, 46, 55, 127, 146, 156] [31, 47, 56, 128, 147, 157] [32, 48, 57, 129, 148, 158] [33, 49, 58, 130, 149, 159] [34, 50, 59, 131, 150, 160] [35, 51, 60, 132, 151, 161] [38, 54, 63, 135, 154, 164] [39, 55, 64, 136, 155, 165] [40, 56, 65, 137, 156, 166] [41, 57, 66, 138, 157, 167] [42, 58, 67, 139, 158, 168] [43, 59, 68, 140, 159, 169] [44, 60, 69, 141, 160, 170] [47, 63, 72, 144, 163, 173] [48, 64, 73, 145, 164, 174] [49, 65, 74, 146, 165, 175] [50, 66, 75, 147, 166, 176] [51, 67, 76, 148, 167, 177] [52, 68, 77, 149, 168, 178] [53, 69, 78, 150, 169, 179] [56, 72, 81, 153, 172, 182] [57, 73, 82, 154, 173, 183] [58, 74, 83, 155, 174, 184] [59, 75, 84, 156, 175, 185] [60, 76, 85, 157, 176, 186] [61, 77, 86, 158, 177, 187] [62, 78, 87, 159, 178, 188] [65, 81, 90, 162, 181, 191] [66, 82, 91, 163, 182, 192] [67, 83, 92, 164, 183, 193] [68, 84, 93, 165, 184, 194] [69, 85, 94, 166, 185, 195] [70, 86, 95, 167, 186, 196] [71, 87, 96, 168, 187, 197] [0, 101] [1, 102] [2, 103] [3, 104] [4, 105] [5, 106] [6, 107] [92, 189] [93, 190] [94, 191] [95, 192] [96, 193] [97, 194] [98, 195] [83, 180] [84, 181] [85, 182] [86, 183] [87, 184] [88, 185] [89, 186] [9, 100, 101, 110] [10, 101, 102, 111] [11, 102, 103, 112] [12, 103, 104, 113] [13, 104, 105, 114] [14, 105, 106, 115] [15, 106, 107, 116] [9, 18, 109, 119] [10, 19, 110, 120] [11, 20, 111, 121] [12, 21, 112, 122] [13, 22, 113, 123] [14, 23, 114, 124] [15, 24, 115, 125] [76, 173, 189, 192] [77, 174, 190, 193] [78, 175, 191, 194] [79, 176, 192, 195] [80, 177, 193, 196] [74, 90, 171, 190] [75, 91, 172, 191] [76, 92, 173, 192] [77, 93, 174, 193] [78, 94, 175, 194] [79, 95, 176, 195] [80, 96, 177, 196]

H_Z (95 checks, sparse supports)

[0, 10, 29, 101, 110, 126] [1, 11, 30, 102, 111, 127] [2, 12, 31, 103, 112, 128] [3, 13, 32, 104, 113, 129] [4, 14, 33, 105, 114, 130] [5, 15, 34, 106, 115, 131] [6, 16, 35, 107, 116, 132] [9, 19, 38, 110, 119, 135] [10, 20, 39, 111, 120, 136] [11, 21, 40, 112, 121, 137] [12, 22, 41, 113, 122, 138] [13, 23, 42, 114, 123, 139] [14, 24, 43, 115, 124, 140] [15, 25, 44, 116, 125, 141] [18, 28, 47, 119, 128, 144] [19, 29, 48, 120, 129, 145] [20, 30, 49, 121, 130, 146] [21, 31, 50, 122, 131, 147] [22, 32, 51, 123, 132, 148] [23, 33, 52, 124, 133, 149] [24, 34, 53, 125, 134, 150] [27, 37, 56, 128, 137, 153] [28, 38, 57, 129, 138, 154] [29, 39, 58, 130, 139, 155] [30, 40, 59, 131, 140, 156] [31, 41, 60, 132, 141, 157] [32, 42, 61, 133, 142, 158] [33, 43, 62, 134, 143, 159] [36, 46, 65, 137, 146, 162] [37, 47, 66, 138, 147, 163] [38, 48, 67, 139, 148, 164] [39, 49, 68, 140, 149, 165] [40, 50, 69, 141, 150, 166] [41, 51, 70, 142, 151, 167] [42, 52, 71, 143, 152, 168] [45, 55, 74, 146, 155, 171] [46, 56, 75, 147, 156, 172] [47, 57, 76, 148, 157, 173] [48, 58, 77, 149, 158, 174] [49, 59, 78, 150, 159, 175] [50, 60, 79, 151, 160, 176] [51, 61, 80, 152, 161, 177] [54, 64, 83, 155, 164, 180] [55, 65, 84, 156, 165, 181] [56, 66, 85, 157, 166, 182] [57, 67, 86, 158, 167, 183] [58, 68, 87, 159, 168, 184] [59, 69, 88, 160, 169, 185] [60, 70, 89, 161, 170, 186] [63, 73, 92, 164, 173, 189] [64, 74, 93, 165, 174, 190] [65, 75, 94, 166, 175, 191] [66, 76, 95, 167, 176, 192] [67, 77, 96, 168, 177, 193] [68, 78, 97, 169, 178, 194] [69, 79, 98, 170, 179, 195] [8, 134] [17, 143] [26, 152] [35, 161] [44, 170] [53, 179] [62, 188] [71, 197] [27, 99, 108] [36, 108, 117] [45, 117, 126] [54, 126, 135] [63, 135, 144] [72, 144, 153] [81, 153, 162] [90, 162, 171] [7, 133, 143] [16, 142, 152] [25, 151, 161] [34, 160, 170] [43, 169, 179] [52, 178, 188] [61, 187, 197] [7, 17, 133] [16, 26, 142] [25, 35, 151] [34, 44, 160] [43, 53, 169] [52, 62, 178] [61, 71, 187] [70, 80, 196] [9, 28, 100, 109] [18, 37, 109, 118] [27, 46, 118, 127] [36, 55, 127, 136] [45, 64, 136, 145] [54, 73, 145, 154] [63, 82, 154, 163] [72, 91, 163, 172]

Code ID 198-12-7 · download JSON · raw on GitHub